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The (first order) flatness of $M$ is shown by using the strongest criterion: $\\{{e_i}\\}$ be an orthonormal basis of $T_{p}M$ and $\\{b_{e_{i}}\\}$ be the corresponding Busemann functions on $M$. Then, (1) The vector space $V = span\\{b_{v} | v \\in T_{p}M \\}$ is finite dimensional and dim $V = $ dim $M = n$.(2) $\\{\\nabla b_{e_i}(p) \\}$ is a global parallel ortho"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1703.00341","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2017-03-01T15:20:49Z","cross_cats_sorted":[],"title_canon_sha256":"331d3c65207821e89fce3ffd0fbb451421b3858a4c02866a0dbec7a820837009","abstract_canon_sha256":"4a0c0778aede8ec7f6c92bd9eb526ee6ba9e5083beff88cfea94afc5ee0547da"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:09.140542Z","signature_b64":"LEBWhSxMBVDmiBOoxfqdNdRJwjWK+lxjXG5FamOuio9E7i58yAJ8hYuSY0/lLvvd+afFuhuLUXO/WCM9Zb9dDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"713ea1e709a0aa6e1d30bb5a2871568191a9fa15d90ea87eee51501f68610fed","last_reissued_at":"2026-05-18T00:23:09.139851Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:09.139851Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Geometry of Asymptotically harmonic manifolds with minimal horospheres","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Hemangi Shah","submitted_at":"2017-03-01T15:20:49Z","abstract_excerpt":"$(M^n,g)$ be a complete Riemannian manifold without conjugate points. 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